One of the pedagogically interesting things about geometry is that there are so many different ways to approach it: one can teach Euclidean or non-Euclidean geometry, with or without linear or abstract algebra, from an analytic or synthetic approach, and can adjust the level of rigor to the needs of a class, from a hyper-rigorous approach where even the most intuitively obvious facts are proved to a more relaxed approach where reasonable inferences from diagrams are allowed. For this reason there are geometry books that approach the subject from many different points of view; a sample of them is provided in the opening paragraphs of the review in this column of Isaacs’ *Geometry for College Students, *and therefore need not be re-catalogued here.

Given this diversity in the literature, it is somewhat surprising that there are not more books that approach the subject from an historical point of view, as does this text. The only other recent texts of which I am aware that also do so are *Revolutions of Geometry* by O’Leary and *Worlds out of Nothing* by Gray, both previously (and favorably) reviewed in this column; the latter, however, is focused on teaching the history of geometry (primarily in the 19^{th} century) rather than on teaching geometry itself. So, that leaves the book under review and *Revolutions* as perhaps the only recent books that teach geometry by following its historical development. However, although the pickings are slim, the fruit is ripe.

Ostermann and Wanner’s book is in two parts. The first, comprising five chapters and entitled “Classical Geometry”, eschews the early pre-history of geometry and begins with its development as a mathematical discipline, with Thales of Miletus in ancient Greece, circa 600 BC. (I cannot resist mentioning that Rex Stout, the late author of the Nero Wolfe mystery novels, got some fictional mileage out of Thales: in the novel *Death of a Doxy*. Wolfe was able to deduce that the murderer was a teacher of mathematics because he signed a blackmail note “Milton Thales”, which of course everybody else pronounced “Thails” rather than “Thay-leeze”. This is, by the way, one of two examples I know of where knowledge of mathematics allowed Wolfe to deduce the identity of the murderer; for the other, see *The Zero Clue*.)

Some results attributed to Thales are proved in the first chapter, which also discusses Pythagoras and his famous theorem, seven different proofs of which are given. Subsequent chapters in Part I take us from Euclid’s *Elements* and Apollonius’s *Conics* through topics such as the theorems of Menalaus and Ceva, the Euler line, the nine-point circle, Steiner circles, Morley’s theorem, etc., some of which are relatively modern. Morley’s theorem, for example, was discovered in 1904; the authors give one proof in the text and another in the exercises.

A final chapter in this first part of the book discusses trigonometry, starting with Ptolemy’s “chord function” that was exclusively used during the Greek period and proceeds through the development of the sine and cosine functions, culminating in a discussion of the work of Kepler and Newton (“don’t expect any easy bedtime reading here”, the authors warn).

Part II of the book has the innocent title “Analytic Geometry” but there is much more going on in the six chapters that comprise this half of the book than the subject with the same name that we all learned about in junior high. The basic theme of part II of the book is an exploration of how geometry has been enriched by the methods of algebra (and, in some chapters, calculus). Accordingly, Part II begins with a discussion (chapter 6) of Descartes’ *Géométrie*, which developed the idea that algebra and geometry could enrich each other, and some specific illustrations of that fact.

The next, fairly long, chapter introduces Cartesian coordinates and provides a variety of interesting applications of them (including identification of various points associated to triangles in terms of coordinates, conic sections, optimization problems in triangles, and many others), and chapter 8 then gives additional applications to ruler and compass constructions, discussing the constructability of the regular n-gon (it is proved that the 17-gon is constructible and the 7-gon is not, and the characterization of those positive integers n for which the regular n-gon is constructible is stated but not proved) and various other impossible constructions (trisecting the angle, doubling the cube, etc.).

The next two chapters introduce vector and matrix methods and multi-dimensional geometry. Here again there are interesting applications, including proofs of Pick’s Theorem (on the area of a simple closed lattice polygon in terms of the number of lattice points in its interior) and a result that I had never heard of before, first proved by van der Waerden in 1970: a pentagon in three-space with all sides and interior angles equal must be planar.

The eleventh and final chapter puts analytic-geometry methods to use to study projective geometry, starting with a discussion of perspective and central projection and proceeding through topics such as the theorems of Desargues, Brianchon and Poncelet, the principle of duality, and projective transformations (with brief reference to Klein’s *Erlanger Programm*). Because the intent of the chapter is to showcase the methods of analytic geometry, the authors do not present an axiomatic development of projective planes but instead limit the discussion to the classical example of extended Euclidean spaces. Homogenous coordinates are defined, but only with real coordinates; thus, for example, the relationship between Pappus’ theorem and the commutativity of the coordinatizing field is not explored; nor is the subject of non-Desarguesian planes.

As this summary should make clear, there is a lot of interesting material in this book, supplemented by a lot of very nice artwork and many interesting exercises (with solutions provided in an Appendix of about 50 pages). There are also (as is inevitable, given the fact that books must be kept to a reasonable length) some significant omissions, most notably with regard to non-Euclidean geometry. The names Bolyai and Lobachevsky are mentioned briefly but there is no real development of their work (although references for further reading are given); by contrast, *Revolutions* spends more than a hundred pages on this topic, as well as some introductory chapters on the axiomatic method, another subject that the authors have chosen not to deal with at any length here. (They do acknowledge the importance of axiom systems but state that “their austere character often discourages beginners”, and point out that Hilbert, in his *Geometry and the* *Imagination*, does not even mention his own set of axioms for Euclidean geometry.) On the other hand, there are also some things in this book (such as the advanced Euclidean geometry of the triangle) that are not covered in any depth in *Revolutions*.

This book arrived at a propitious time for me, because during the next academic year I will be teaching a two-semester sequence on Euclidean and non-Euclidean geometry. Because the topics in the course don’t quite align with the topics in this book, I won’t be able to use this book as a text for the course, but it is a pretty safe bet that I will keep it close at hand and refer to it often for additional ideas. I would think that any other college instructor (or student, for that matter) with an interest in geometry would also want a copy on his or her shelf.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.